I. INTRODUCTION
The problem of plasma expansion into vacuum has been studied for many years since the paper of Gurevich et al. (Reference Gurevich, PariÏskaya and PitaevskiÏ1966). Interest in this problem is crucial for better understanding the physics of ion acceleration in the laser–plasma interaction and in particular, to give a quantitative description of this phenomenon. The study of ion acceleration is among the key problems in various applications of high-power lasers, such as laser fusion, injectors of fast particles, and radioactive sources used in medicine and nuclear physics (Kovalev et al., Reference Kovalev, Bychenkov and Tikhonchuk2001).
Most studies of plasma expansion into vacuum are based on a semi-bounded plasma model with isothermal electrons and cold ions (Wickens et al., Reference Wickens, Allen and Rumsby1978; Gurevich et al., Reference Gurevich, Anderson and Wilhelmsson1979; Mora & Pellat, Reference Mora and Pellat1979; Gurevich & Meshcherkin, Reference Gurevich and Meshcherkin1981). It is however proven from experiments (Bulgakova et al., Reference Bulgakova, Bulgakov and Bobrenok2000) that the hypothesis of isothermal electrons is not appropriate for laser–plasma interactions. A more realistic model has to account for two-electron populations: Cold dominant and hot minority. It was shown in the literature, for example in (Breizman & Arefiev, Reference Breizman and Arefiev2007) that, in contrast to the two-component distribution, a single-temperature distribution of electrons is predestined to underestimate the energy of accelerated ions for a given absorbed laser power. In laser produced plasma, the deviation from Maxwell distribution, at the earlier stage, is attributed to the existence of an electron emission relaxation region. In this region the plasma interaction with matter can be modeled by assuming a partially ionized gas separated from the surface by an electrical sheath in which charged particles are accelerated. Thus, two groups of electrons exist; accelerated electrons emitted from the heated target and slow plasma electrons. This leads to a collisional dominated plasma heated by accelerated electrons (Beilis, Reference Beilis2007). With moderate laser energies (106−1010 W/cm2), ion acceleration mechanism is explained by invoking the shifted Maxwellian ion energy distribution (Beilis, Reference Beilis2012). It is also assumed, as a consequence of these two groups of electrons, that increasing laser intensity as well as energy causes an increase in emission of both fast (hot) and thermal (cold) ions (Wolowski et al., Reference Wołowski, Celona, Ciavola, Gammino, Krása, Láska, Parys, Rohlena, Torrisi and Woryna2002; Krasa et al., Reference Krasa, Jungwirth, Krousky, Laska, Rohlena, Pfeifer, Ullschmied and Velyhan2007). The fast electrons component escaping from the interaction region has been observed and interpreted as a signature of a two-electron temperature plasma formation (Mascali et al., Reference Mascali, Tudisco, Gambino, Pluchino, Anzalone, Musumeci, Rapisarda and Spitaleri2012). There, a relatively small number of hot electrons, accelerated by laser, enters into a cold target and induces the ion acceleration. The electron density could then be considered as a sum of two Boltzmann distributions with cold (T ec) and hot (T eh) temperatures (Tikhonchuk et al., Reference Tikhonchuk, Andreev, Bochkarev and Bychenkov2005; Diaw & Mora, Reference Diaw and Mora2011) given by:
$${n_{\rm e}} = {n_{{\rm ec}}} + {n_{{\rm eh}}} = {n_{{\rm ec0}}}{\rm exp}\left( {e{\rm \varphi} /{T_{{\rm ec}}}} \right) + {n_{{\rm eh}0}}{\rm exp}\left( {e{\rm \varphi} /{T_{{\rm eh}}}} \right),$$with T ec ≪ T eh and n ec0 ≫ n eh0. n ec0, n eh0 are the cold and hot initial densities, respectively, and φ is the electric potential.
On the other hand, it was proven by many authors that fast electrons have non-Maxwellian distributions in the laboratory experiments, for example, during high-intensity laser–matter interactions, where the generated plasma fails to thermalize within the short observation time interval (Hellberg et al., Reference Hellberg, Mace, Armstrong and Karlstad2000; Yoon et al., Reference Yoon, Rhee and Ryu2005; Goldman et al., Reference Goldman, Newman and Mangeney2007; Sarri et al., Reference Sarri, Dieckmann, Kourakis and Borghesi2010). The fundamental reason is that fast electrons collide much less frequently than slow ones. Indeed their free path is very large since proportional to 
$v_{\rm e}^4 $, where v e is the electron velocity, and cannot relax to a Maxwellian (Bennaceur-Doumaz & Djebli, Reference Bennaceur-Doumaz and Djebli2010). Moreover, measurements show that the electron distribution function in the laser-pellet experiments is a non-Maxwellian with an energetic tail. These energetic electrons could effectively have a significant effect on ionization and expansion processes (Gurevich et al., Reference Gurevich, Anderson and Wilhelmsson1979; Yoon et al., Reference Yoon, Rhee and Ryu2005; Beilis, Reference Beilis2012). It was also found that the moving space-charge electric field accelerates ions to energies well above the energy of the fast electrons (Mora, Reference Mora2003, Reference Mora2005). A quantitative comparison with existing theories involving only Maxwellian plasmas is then not accurate since in the experiment the energetic electrons have a non-Maxwellian distribution (Hairapetian & Stenzel, Reference Hairapetian and Stenzel1988, Reference Hairapetian and Stenzel1991). Various models have been proposed to analyze the qualitative behavior of plasma expansion assuming, for instance, for the fast electrons, different non-Maxwellian distribution functions such as a truncated Maxwellian distribution (Lontano & Passoni, Reference Lontano and Passoni2006), a super-Gaussian distribution (Kovalev et al., Reference Kovalev, Bychenkov and Tikhonchuk2001, Reference Kovalev, Bychenkov and Tikhonchuk2002), Cairns distribution (Bennaceur-Doumaz & Djebli, Reference Bennaceur-Doumaz and Djebli2010), a steplike electron energy distribution, (Kiefer et al., Reference Kiefer, Schlege and Kaluza2013) or kappa distribution (Hellberg et al., Reference Hellberg, Mace, Armstrong and Karlstad2000).
In this paper, we proposed for the first time, in the framework of plasma expansion and ion acceleration, a theoretical model for the free expansion of semi-infinite plasma into vacuum combining cold Maxwellian electrons with hot ones assumed to be non-Maxwellian. For the latter, the electrons are supposed to follow kappa distribution function. This distribution function has a Maxwellian-like core and a high-energy component of power-law form, which reproduces smoothly the velocity dependence. In laser produced plasma, it has been already used to model ion acceleration and plasma expansion for one-electron temperature plasma in (Shokoohi & Abbasi, Reference Shokoohi and Abbasi2009; Mehdian et al., Reference Mehdian, Kargarian and Hajisharifi2014; Bennaceur-Doumaz et al., Reference Bennaceur-Doumaz, Bara, Benkhelifa and Djebli2015), where the authors showed numerically that by increasing the population of energetic electrons, the expansion took place faster, the resulting electric field was stronger, and the ions accelerated to higher energy.
2. BASIC EQUATIONS OF THE MODEL
2.1. Electron density
The present work concerns the study of one-dimensional (1D), non-relativistic, collisionless expansion into a vacuum of semi-infinite laser produced plasma. The approximation of a 1D description of the expansion is justified as long as lateral heat conduction can be neglected. In order to describe the realistic situation we have to include the cold electrons in the plasma, because not all of the target electrons are heated up by the laser, so the plasma is considered with two-electron populations, hot electrons and cold electrons with much lower temperature, but higher density.
The cold population follows Boltzmann distribution while the hot one, which can be considered as fast, is governed by the kappa distribution. The total electron density is then the sum of the cold and hot electron densities given by:
$$\eqalign{{n_{\rm e}} & = {n_{{\rm ec}}} + {n_{{\rm eh}}} \cr & = {n_{{\rm ec}0}}{\rm exp}\left( {e{\rm \varphi} /{T_{{\rm ec}}}} \right) + {n_{{\rm eh}0}}{\left( {1 - \displaystyle{{e{\rm \varphi} /{T_{{\rm eh}}}} \over {{\rm (\kappa} - 3/2{\rm )}}}} \right)^{ - {\rm \kappa} + 1/2}}}$$n e0 = n ec0 + n eh0 is the total initial density, κ ≥ 3/2 is the spectral index which measures the strength of the excess superthermality. When κ → ∞, we retrieve the case of two-electron temperature Maxwellian plasma.
2.2 Ion motion modeling
The cold ion motion is modeled by fluid equations:
$$\displaystyle{{\partial {n_{\rm i}}} \over {\partial t}} + {v_{\rm i}}\displaystyle{{\partial {\rm (}{n_{\rm i}}{v_{\rm i}}{\rm )}} \over {\partial x}} = 0$$
$$\displaystyle{{\partial {v_{\rm i}}} \over {\partial t}} + {v_{\rm i}}\displaystyle{{\partial {v_{\rm i}}} \over {\partial x}} + \displaystyle{{Ze} \over {{m_{\rm i}}}}\displaystyle{{\partial {\rm \varphi}} \over {\partial x}} = 0,$$where n i, v i, and m i are the ion density, ion velocity, and ion mass, respectively. We assume a plasma with single charged ions Z = 1.
Electron–electron collisions which would eventually equilibrate the hot and cold species are ignored; electron–ion collisions are also neglected (True et al., Reference True, Albritton and Williams1981).
2.3. Quasi-neutrality of charge
During the expansion of hot plasma under vacuum, a sizable fraction of the beam energy is carried by the quasi-neutral region of the expanding plasma and the characteristic scale length for plasma density variations is generally large compared with the Debye length, so that the plasma remains quasi-neutral during all the expansion. The equation of quasi-neutrality is given by:
$${n_{\rm e}} = {n_{{\rm ec}}} + {n_{{\rm eh}}} = {n_{\rm i}}$$The quasi-neutrality condition guarantees the validity of self-similar solutions, motivating the interest of the present paper in these types of solutions.
3. SELF-SIMILAR SOLUTION
In the presence of free boundary associated with plasma expansion, under certain assumptions, the partial differential Eqs (2) and (3) can be reduced to ordinary differential equations. This transformation is based on the assumption that we have a self-similar solution, that is, every physical parameter distribution preserves its shape during expansion and there is no scaling parameter (Zel'dovich & Raizer, Reference Zel'dovich and Raizer1966; Sack & Schamel, Reference Sack and Schamel1987). Self-similar solutions usually describe the asymptotic behavior of an unbounded problem and the time t and the space coordinate x appear only in the combination of τ = x/t. It means that the existence of self-similar variables implies the lack of characteristic lengths and times. Indeed, this is justified when we deal with the assumption of charge quasi-neutrality in the plasma. The Debye length then, loses its importance as a characteristic length in the plasma.
In the framework of self-similar theory, we obtain the following set of ordinary equations:
$$\left( {{v_{\rm i}} - {\rm \tau}} \right)\displaystyle{{d{n_{\rm i}}} \over {d{\rm \tau}}} + {n_{\rm i}}\displaystyle{{d{v_{\rm i}}} \over {d{\rm \tau}}} = 0$$
$$\left( {{v_{\rm i}} - {\rm \tau}} \right)\displaystyle{{d{v_{\rm i}}} \over {d{\rm \tau}}} + \displaystyle{e \over {{m_{\rm i}}}}\displaystyle{{d{\rm \varphi}} \over {d{\rm \tau}}} = 0$$Deriving (1) with respect to τ and using the equation of quasi-neutrality (4), we obtain,
$$\displaystyle{{d{n_{\rm e}}} \over {d{\rm \tau}}} = \displaystyle{{d{n_{\rm i}}} \over {d{\rm \tau}}} = Ae\displaystyle{{d{\rm \varphi}} \over {d{\rm \tau}}} $$
$$\eqalign{{\rm A} & = \displaystyle{{{n_{{\rm ec}0}}} \over {{T_{{\rm ec}}}}}{\rm exp}\left( {e{\rm \varphi} /{T_{{\rm ec}}}} \right) \cr & + \displaystyle{{{n_{{\rm eh}0}}} \over {{T_{{\rm eh}}}}}\displaystyle{{{\rm (\kappa} - {\rm 0}{\rm. 5)}} \over {{\rm (\kappa} - {\rm 1}{\rm. 5)}}}{\left( {1 - \displaystyle{{e{\rm \varphi} /{T_{{\rm eh}}}} \over {{\rm (\kappa} - {\rm 1}{\rm. 5)}}}} \right)^{\left( { - {\rm \kappa} - 0.5} \right)}} \cr & = \displaystyle{{{n_{{\rm ec}}}} \over {{T_{{\rm ec}}}}} + \displaystyle{{{n_{{\rm eh}0}}} \over {{T_{{\rm eh}}}}}\displaystyle{{{\rm (\kappa} - {\rm 0}{\rm. 5)}} \over {{\rm (\kappa} - {\rm 1}{\rm. 5)}}}{\left( {\displaystyle{{{n_{{\rm eh}}}} \over {{n_{{\rm eh}0}}}}} \right)^{\left( { - {\rm \kappa} - 0.5} \right)/\left( { - {\rm \kappa} + 0.5} \right)}}}$$Equation (6) becomes:
$$\displaystyle{1 \over {A{m_{\rm i}}}}\displaystyle{{d{n_{\rm i}}} \over {d{\rm \tau}}} + \left( {{v_{\rm i}} - {\rm \tau}} \right)\displaystyle{{d{v_{\rm i}}} \over {d{\rm \tau}}} = 0$$If we treat all the derivative terms as independent variables and the resulting set of equations as algebraic ones, then the non-trivial solution to the system of Eqs (5) and (9) requires that the determinant of their coefficients must vanish, that is, while choosing the positive solution corresponding to an expansion in the +x direction and a velocity increasing with increasing x (Yu & Luo, Reference Yu and Luo1995):
$$\eqalign{{v_{\rm i}} - {\rm \tau} & = \sqrt {\displaystyle{{{n_{\rm i}}} \over {{m_{\rm i}}A}}} \cr & = \sqrt {\displaystyle{{{n_{\rm i}}} \over {{m_{\rm i}}\left( \matrix{{n_{{\rm ec}}}/{T_{{\rm ec}}} + {n_{{\rm eh}0}}/{T_{{\rm eh}}}{\rm (\kappa} - {\rm 0}{\rm. 5)}/ \hfill \cr \quad {\rm (\kappa} - {\rm 1}{\rm. 5)(}{n_{{\rm eh}}}/{n_{{\rm eh}0}}{{\rm )}^{\left( { - {\rm \kappa} - 0.5} \right)/\left( { - {\rm \kappa} + 0.5} \right)}} \hfill} \right)}}} = {c_{\rm s}}}$$c s is the ion sonic velocity of the plasma.
The metal target used in producing laser-plasma is supposed unlimited and is homogeneously filled negative half space. At time t = 0, the target surface involved in the experiments is located at x = 0, and the target is in x < 0 region. Laser radiation is switched on in –x direction and the plasma starts expanding. Ions are cold and initially at rest with density, n i = n i0 for x < 0 and n i = 0 for x > 0 with a sharp boundary. At t > 0, plasma begins to expand into vacuum. Due to the expansion, the plasma density will decrease and hence a rarefaction wave will propagate in the –x direction (Cheng et al., Reference Cheng, Perrie, Wub, Tao, Edwardson, Dearden and Watkins2009) with the velocity c s.
Differentiating (10) and using (8) gives:
$$\displaystyle{{d{v_{\rm i}}} \over {d{\rm \tau}}} = 1 + 0.5\sqrt {\displaystyle{1 \over {{m_{\rm i}}}}} \left( {\displaystyle{{e\sqrt A} \over {\sqrt {{n_{\rm i}}}}} - \displaystyle{{\sqrt {{n_{\rm i}}}} \over {{A^{1.5}}}}B} \right)\displaystyle{{d{\rm \varphi}} \over {d{\rm \tau}}} {\rm,} $$where
$$\displaystyle{{dA} \over {d{\rm \tau}}} = Be\displaystyle{{d{\rm \varphi}} \over {d{\rm \tau}}} $$and
$$\eqalign{& {B =\, } \left( {\displaystyle{{{{n}_{{\rm ec}0}}} \over {{T}_{{\rm ec}}^{\rm 2}}}} \right){\rm exp}\left( {{e}{\rm \varphi} /{{T}_{{\rm ec}}}} \right){\rm +} \left( {\displaystyle{{{{n}_{{\rm eh}0}}} \over {{T}_{{\rm eh}}^2}}} \right) \cr & \qquad\times \displaystyle{{{\rm (\kappa} - {\rm 0}{\rm. 5)(\kappa} + {\rm 0}{\rm. 5)}} \over {{{{\rm (\kappa} - {\rm 1}{\rm. 5)}}^2}}}{\left( {1 - \displaystyle{{{e}{\rm \varphi} /{{T}_{{\rm eh}}}} \over {{\rm (\kappa} - {\rm 1}{\rm. 5)}}}} \right)^{{\rm (} - {\rm \kappa} - 1.5{\rm )}}}} $$Using Eqs (6), (10), and (11), we found the equation to solve for the electrostatic potential as:
$$e\displaystyle{{d{\rm \varphi}} \over {d{\rm \tau}}} = - \displaystyle{{\sqrt {{m_{\rm i}}} \sqrt {{n_{\rm i}}}} \over {1.5\sqrt A - 0.5{n_{\rm i}}B/{A^{1.5}}}}$$Potential, velocity, and density of the ions are given numerically by the solution of the system of Eqs (12), (7), and (11) under the assumption of quasi-neutrality using Runge–Kutta method.
The expansion profiles are deduced in normalized forms such as 
${\tilde n_{\rm i}} = {n_{\rm i}}/{n_{{\rm i}0}}$, 
${\tilde v_{\rm i}} = {v_{\rm i}}/{c_{{\rm sh}}}$ and Φ = eφ/T eh, according to the dimensionless self-similar variable ξ = x/c sht = τ/c sh, where 
${c_{{\rm sh}}} = \sqrt {{T_{{\rm eh}}}/{m_{\rm i}}} $ is the ion sound velocity in the hot electron species.
The initial time t = 0 in our case corresponds to an unperturbed plasma with initial parameters 
${\tilde n_{\rm i}} = 1$ and 
${\tilde v_{\rm i}} = 0$ (Ivlev & Fortov, Reference Ivlev and Fortov1999; Bara et al., Reference Bara, Djebli and Bennaceur-Doumaz2014). As a consequence, we require that there should exist a point ξ0 at t ≤ 0 for which the plasma is unperturbed and at rest, such that 
${\tilde v_{\rm i}}{\rm (}{{\rm \xi} _0}{\rm )} = 0$, 
${\tilde n_{\rm i}}{\rm (}{{\rm \xi} _0}{\rm )} = 1$, and Φ(ξ0) = 0. From Eq. (10), its expression is given by:
$${{\rm \xi} _0} = - \sqrt {{{\left( {\displaystyle{{\rm \alpha} \over {{\rm (}1 + {y_{\rm u}}{\rm )}}} + \displaystyle{{{y_{\rm u}}} \over {{\rm (}1 + {y_{\rm u}}{\rm )}}}\displaystyle{{{\rm (\kappa} - 0.5{\rm )}} \over {{\rm (\kappa} - 1.5{\rm )}}}} \right)}^{ - 1}}}, $$where y u = n eh0/n ec0 and α = T eh/T ec.
4. DISCUSSION OF THE RESULTS
4.1. Case of two-electron Maxwellian populations
Bezzerides et al. (Reference Bezzerides, Forslund and Lindman1978) have shown that for the case of two-electron Maxwellian distributions, a continuous self-similar solution for semi-infinite expanding plasma by assuming the quasi-neutrality is limited when the solution for the potential φ(x/t) is multi-valued, that is, when temperature ratio α is higher than 
$5 + \sqrt {24} \approx 9.9$. In the point of view of fluid theory, it means that the solutions of these equations become double valued when 
$dc_{\rm s}^2 /d{\rm \varphi} + 2{\rm \lt 0}$ where c s is the ion sound velocity defined by (Diaw & Mora, Reference Diaw and Mora2011):
$$c_{\rm s}^2 = \displaystyle{{{\rm (}{n_{{\rm ec}}} + {n_{{\rm eh}}}{\rm )}} \over {{m_{\rm i}}{\rm (}{n_{{\rm ec}}}/{T_{{\rm ec}}} + {n_{{\rm eh}}}/{T_{{\rm eh}}}{\rm )}}} = c_{{\rm sh}}^2 \left( {\displaystyle{{1 + y} \over {{\rm \alpha} + y}}} \right)$$obtained from Eq. (10) when κ → ∞ and y = n eh/n ec is the density ratio.
Since double-valued solutions are unphysical, they become discontinuous under such conditions. For higher temperature ratios, they have deduced that a collisionless shock front in the rarefaction wave is formed, and it provides a spatial separation of the two-electron populations. Then, ions are accelerated at the shock front to the hot ion acoustic velocity.
To show the influence of two-electron temperatures on the plasma expansion, we present, in Figures 1 and 2, normalized ion densities and normalized ion velocities, respectively, as functions of the self-similar variable ξ for different values of the parameter α up to the α limit value for a given density ratio y u.
Fig. 1. Variation of the normalized ion density with ξ for different values of temperature ratios α, for initial density ratio y u = 0.1.
Fig. 2. Variation of the normalized ion velocity with ξ for different values of temperature ratios α, for initial density ratio y u = 0.1.
In Figure 1, the self-similar solutions show two behaviors of ion density relatively to the intersection point ξ = 0, considered as the original position of the plasma surface. The different ion profiles are compared with the one-electron temperature plasma expansion represented by α = 1, published for the first time by Gurevich et al. (Reference Gurevich, PariÏskaya and PitaevskiÏ1966).
Near the source, ξ < 0, first the hot electrons leave the plasma, pulling the ions behind. During the expansion of plasma, space-charge effects self-consistently produce an ambipolar electric field whose amplitude is controlled by the energy of the hot electrons. The ambipolar electric field accelerates a small number of ions to streaming energies. The thermal electrons follow, guiding the slower thermal ion group (Rohlena et al., Reference Rohlena, Kralikova, Krasa, Laska, Masek, Pfeifer, Skala, Parys, Wolowski, Woryna, Farny, Mroz, Roudskoyji, Shamaev, Sharkov, Shumshurov, Bryunetkin, Haseroth, Collier, Kuttenbeger, Langbein and Kugler1996). The ionic sound velocity in this region is given from Eq. (14) by 
${c_{\rm s}} \approx {c_{{\rm sh}}}\sqrt {{\rm (}1/{\rm \alpha )}} = {c_{{\rm sc}}} = \sqrt {{T_{{\rm ec}}}/{m_{\rm i}}} $, which is the ion sound velocity of the cold electrons species only, since there y ≪ 1. Consequently, it is the cold electron population which governs the expansion; indeed, the expansion is slowing down relatively to the isothermal case α = 1 and the ion depletion is less pronounced. The density is given for the cold electron plasma as 
${\tilde n_{\rm i}} \approx {\rm exp(} \! - \sqrt {\rm \alpha} {\rm \xi} - 1{\rm )}$.
Far from the target, a visible drop of density is shown. The ion depletion is more pronounced with increasing temperature ratio, indicating that ion acceleration is more effective and driven by the hot electrons, whereas majority of cold electrons are retarded by the self-consistent electric field. As the expansion progresses, the potential becomes more negative and the hotter electrons can no longer be regarded as a low number density component. In fact, at sufficiently negative potentials they become the majority species, and might then be expected to have the dominant effect on the ion acceleration. The ionic sound velocity is then given by c s ≈ c sh from Eq. (14) because there y ≫ 1 and the density tends to the asymptote 
${\tilde n_{\rm i}} \approx {\rm exp}\left( { - {\rm \xi} - 1} \right)$ at the ion front (Wickens & Allen, Reference Wickens and Allen1979). At α = 9.9, a discontinuity begins to appear clealy indicating the separation of the two-electron populations and the onset of the shock front. Moreover, it is also pointed out that the end of the self-similar expansion corresponds to vanishing density.
In Figure 2, we have plotted the ion velocity profiles with different values of α. For α = 1, as it is well-known, the self-similar ion velocity is linear for the one temperature case (Gurevich et al., Reference Gurevich, PariÏskaya and PitaevskiÏ1966). However, for α ≠ 1, the figures show an expansion with three phases:
• a zone of slow expansion dominated by cold electrons and due to charge quasi-neutrality, the ions follow the cold electrons and are also slowed with subsonic velocities. The ionic sound velocity is given by c sc and ions move with a linear velocity depending on α parameter such as
${\tilde v_{\rm i}} \approx {\rm \xi} + {\rm (}{c_{{\rm sc}}}/{c_{{\rm sh}}}{\rm )} = {\rm \xi} + \sqrt {1/{\rm \alpha}} $. During the early stages of the expansion the ion acceleration is thus determined by the colder electron species.• an intermediate zone where the two populations coexist and begin to separate. A double layer-like develops across the region where the separation occurs similar to the theoretically predicted rarefaction shock when α tends to its critical value.
• a zone where the expansion is held by the hot electrons ending at the ion front. There, ions are accelerated by strong electric fields to supersonic velocities and experience a strong increase in their energy. The ionic sound velocity tends to c sh such that the ions move linearly with
${\tilde v_{\rm i}} \approx {\rm \xi} + 1$, as in the case of a one-electron temperature plasma (Gurevich & Meshcherkin, Reference Gurevich and Meshcherkin1981; Mora, Reference Mora2003, Reference Mora2005).
The effect of the cold electrons is then restricted to the plasma surface, and a singularity occurs in the potential-profile for high temperature ratios when Eq. (12) vanishes (Bezzerides et al., Reference Bezzerides, Forslund and Lindman1978). At the ion front, there is no big difference between the one- and two-temperature plasma expansion, the velocity is mostly defined by the hot electron parameters, showing that the presence of cold electrons increases very slightly the ion acceleration (e.g., at ξ = 6, for α = 9.9, the increase of velocity is 3.73% relatively to the one temperature case) and the effect of the temperature is small.
To show the influence of density ratios y u on the plasma expansion, we have drawn in Figures 3 and 4, the density and velocity profiles, for different values of y u for a given temperature ratio α. For the densities as it is shown in Figure 3, there is practically no influence of density ratio on the profiles in early plasma expansion where the density tends to the asymptote 
${\tilde n_{\rm i}} \approx {\rm exp} (- \sqrt {\rm \alpha} {\rm \xi} - 1{\rm )}$ but it is seen to have some significance in later expansion where the density is given by 
${\tilde n_{\rm i}} \approx {\rm exp} (- {\rm \xi} - 1 + B{\rm )}$, where
$$B = \displaystyle{1 \over {{\rm (\alpha} - 1{\rm )}}}\left\{ {{{\rm \alpha} ^{0.5}}Ln\left( {\displaystyle{{4y_{\rm u}^{ - 1}} \over {{{{\rm [}1 + {{\rm \alpha} ^{ - 0.5}}{\rm ]}}^2}}}} \right) + Ln\left( {\displaystyle{{4{y_{\rm u}}} \over {{{{\rm [}1 + {{\rm \alpha} ^{0.5}}{\rm ]}}^2}}}} \right)} \right\}$$is a function of y u, as calculated by Wickens and Allen (Reference Wickens and Allen1979). Then for a given temperature ratio, the increasing of cold population relatively to the hot one has the role of enhancing the ion depletion and accelerating the expansion.
Fig. 3. Variation of the normalized ion density with ξ for different values of initial density ratios y u for α = 8.
Fig. 4. Variation of the normalized ion velocity with ξ for different values of initial density ratios y u for α = 8.
In Figure 4, for velocities, we observe the same phenomenon for early plasma expansion where again there is no influence of y u on the profiles since their behavior tends to the asymptote 
${\tilde v_{\rm i}} \approx {\rm \xi} + \left( {{c_{{\rm sc}}}/{c_{{\rm sh}}}} \right) = {\rm \xi} + \sqrt {1/{\rm \alpha}} $, but as the expansion proceeds, the influence of y u is apparent at the ion front where the velocities tend to 
${\tilde v_{\rm i}} \approx {\rm \xi} + B$ (Wickens & Allen, Reference Wickens and Allen1979) and where ion acceleration is more effective with increasing electron cold population.
For example, the contribution of the density ratio to ion acceleration is about 8.13% at ξ = 4.8 when passing from y u = 0.1 to y u = 0.01.
4.2. Effect of non-Maxwellian electron population on the plasma expansion
In this part, we investigate the role of non-Maxwellian electrons on plasma expansion and ion acceleration. In Figure 5, the ion velocity profiles are drawn for different values of κ parameter for α = 9 and y u = 0.1 and compared with the case of two-electron Maxwellian plasma obtained when κ → ∞ in Eq. (1).
Fig. 5. Influence of non-Maxwellian electrons on ion velocity expansion for α = 9 and y u = 0.1.
In early stage of plasma expansion near the target, the ion acoustic velocity is given approximately by the ion acoustic velocity in the cold electrons species that is:
$$\eqalign{ {c_{\rm s}} &= {c_{{\rm sh}}}\sqrt {\displaystyle{{1 + y} \over \matrix{\left( {{n_{{\rm eh}0}}/{n_{{\rm ec}}}{\rm (\kappa - 0}{\rm. 5)}/} \right. \hfill \cr \left. {\quad {\rm (\kappa} - {\rm 1}{\rm. 5)(}{n_{{\rm eh}}}/{n_{{\rm eh}0}}{{\rm )}^{{\rm (} - {\rm \kappa} - 0.5{\rm )/(} - {\rm \kappa} + 0.5{\rm )}}}{\rm +} {\rm \alpha}} \right) \hfill}}} \cr &\approx {c_{{\rm sh}}}\sqrt {\displaystyle{1 \over {\rm \alpha}}} = {c_{{\rm sc}}},}$$
the part dependent of κ is negligible relatively to α so it appears that energetic electrons again have no effect on ion expansion and consequently, in this region, ions move with a linear velocity depending on α parameter such as 
${\tilde v_{\rm i}} \approx {\rm \xi} \,+$
$\left( {{c_{{\rm sc}}}/{c_{{\rm sh}}}} \right) = {\rm \xi} + \sqrt {1/{\rm \alpha}} $.
The role of non-Maxwellian electrons is more apparent beyond the trapping region of the cold electrons and is predominant in the expansion front where their velocity tends to the asymptote 
${\tilde v_{\rm i}} = {\rm (\kappa} - 1/2{\rm )}/{\rm (\kappa} - 1{\rm )\xi}\, +$
$\sqrt {\left( {{\rm \kappa} - 1/2} \right){\rm (\kappa} - 3/2{\rm )}} /{\rm (\kappa} - 1{\rm )} $ already given in (Bennaceur-Doumaz et al., Reference Bennaceur-Doumaz, Bara, Benkhelifa and Djebli2015), where only one-component non-thermal electrons are involved in the expansion. When κ → ∞, the velocity tends to Maxwellian case where 
${\tilde v_{\rm i}} \approx {\rm \xi} + 1$.
We can deduce that the presence of energetic tail electrons whose population enhances with decreasing κ parameter, has the role to increase ion acceleration much more than the acceleration driven by the hot Maxwellian electrons obtained when κ → ∞. Then, supposing two-electron Maxwellian populations induces an underestimation of ion acceleration in laser produced plasma expansion.
It is also worth to indicate that in the present study of ion acceleration, the continuous self-similar solution limit, assuming charge neutrality is increasing when κ is decreasing (increasing of population of non-thermal electrons), reaching the value of 15.9 for κ = 2. This is illustrated in Figure 6 where ion velocities are drawn for different α limits depending on κ parameter. These limit values are obtained numerically when handling the present electron distribution function [Eq. (7)], in contrast with the case of two-electron Maxwellian distributions where the limit α = 9.9 is obtained analytically by Bezzerides et al. (Reference Bezzerides, Forslund and Lindman1978).
Fig. 6. Ion velocities in the α limit of self-similar solution for different values of κ, y u = 0.1.
5. CONCLUSION
The expansion of laser produced plasma is studied in the presence of two-electron temperature populations: A cold one modeled by Maxwellian distribution and a hot one which contains energetic non-Maxwellian electrons modeled by kappa distribution function.
The self-similar solution, obtained shows that the effect of cold electrons is restricted in a region confined in the early stage of expansion whereas the hot ones are responsible of ion acceleration at the ion front. It is found that in addition to thermal effects, the presence of non-Maxwellian electrons enhances the ion expansion and also extends the validity of the self-similar solution. This study could be useful in modeling laser produced plasma expansion in laser ablation experiments for fusion purposes where the intensity of the laser does not exceed 1013 W/cm2 and the α parameter is not too high in order to keep quasi-neutrality condition and self-similar solutions valid during expansion which is not the case when very large electric fields are built with very intense lasers (Tikhonchuk et al., Reference Tikhonchuk, Andreev, Bochkarev and Bychenkov2005; Diaw & Mora, Reference Diaw and Mora2011; Bennaceur-Doumaz et al., Reference Bennaceur-Doumaz, Bara, Benkhelifa and Djebli2015).
